Existence of global solutions for systems of reaction-diffusion equations on unbounded domains

Badraoui, Salah

Electronic Journal of Differential Equations, Tome 2002 (2002).

Résumé

Summary: We consider, an initial-value problem for the thermal-diffusive combustion system $$\displaylines{ u_t=a\Delta u-uh(v) \cr v_t=b\Delta u+d\Delta v+uh(v), }$$ where $d positive, b\neq 0, x\in \mathbb{R}^n, n\geq 1$, with $h(v)=v^m, m$ is an even nonnegative integer, and the initial data $u_0, v_0$ are bounded uniformly continuous and nonnegative. It is known that by a simple comparison if $b=0, a=1, d\leq 1$ and $h(v)=v^m$ with $m\in \mathbb{N}^*$, the solutions are uniformly bounded in time. When $d greater than a=1, b=0, h(v)=v^m$ with $m\in \mathbb{N}^*$, Collet and Xin [2] proved the existence of global classical solutions and showed that the $L^\infty $ norm of $v$ can not grow faster than $O(\log\log t)$ for any space dimension. In our case, no comparison principle seems to apply. Nevertheless using techniques form [2], we essentially prove the existence of global classical solutions if $a less than d, b less than 0$, and $v_0\geq \frac b{a-d}u_0$.

Détail

Classification : 35B40, 35B45, 35K55, 35K65
Keywords: reaction-diffusion systems, positivity, global existence, boundedness